A primer on Gravitational Lensing200.145.112.249/webcast/files/Lensing_DarkMatter_IFT.pdf · A...

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

A primer on Gravitational Lensing

Eduardo S. Cypriano1

1Departamento de AstronomiaInstituto de Astronomia, Geofısica e Ciencias Atmosfericas

Universidade de Sao Paulo

IFT School on Dark Mater, 2016

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

Outline

Gravitational Lensing basics

Micro lensing

Strong lensing

Weak Lensing

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

Gravitational lensing geometry

Figure from Narayan & Bartelmann (1996; arXiv:astro-ph/9606001)

I Parameters:I Angular diameter distances: Ds , Dd and Dds

I Source position: β = η/Ds - Impact parameterI Image position: θ = ξ/Dd

I Deflexion angle: α = α Dds/Ds

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

The Lensing Equation

I

η =Ds

Ddξ − Ddsα(ξ)

I Using the angular quantities instead of the physical oneswe get:

β = θ − Dds

Dsα(Ddθ)

I Then by using the physical deflexion angle we arrive tothe Lens Equation:

β = θ −α(θ)

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

The Lensing Equation

I

η =Ds

Ddξ − Ddsα(ξ)

I Using the angular quantities instead of the physical oneswe get:

β = θ − Dds

Dsα(Ddθ)

I Then by using the physical deflexion angle we arrive tothe Lens Equation:

β = θ −α(θ)

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

The Lensing Equation

I

η =Ds

Ddξ − Ddsα(ξ)

I Using the angular quantities instead of the physical oneswe get:

β = θ − Dds

Dsα(Ddθ)

I Then by using the physical deflexion angle we arrive tothe Lens Equation:

β = θ −α(θ)

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

The deflection angle

I From GR one can estimate the deflection angle due to amass point:

α =4 G M

c2ξ

I The projected angle is then:

α = |α| =Dds

Ds

4 G M

c2Dd |θ|

I Considering now a direction, as all the angles lies in theimage-lens direction, we have:

α =Dds

Ds

4 G M

c2Dd |θ|θ

|θ|

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

The deflection angleI This prediction lead to the first observational proof of

the General Relativity

ANNOUNCEMENTOFEDDINGTON’SDISCOVERY.CREDIT:ILLUSTRATEDLONDON NEWS(1919)

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

The lensing effect

I Gravitational lensing causes several effects on theimages of the sources:

1. Radial displacement

2. Multiple imaging (given certain conditions)3. Magnification of the angular size

surface brightness conservation−−−−−−−−−−−−−−−−−−→ and flux

4. Distortion5. Time Delay

I The prevalence/interest of one or more of those effectsover the others have to do with the lensing regime.

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

The lensing effect

I Gravitational lensing causes several effects on theimages of the sources:

1. Radial displacement2. Multiple imaging (given certain conditions)

3. Magnification of the angular size

surface brightness conservation−−−−−−−−−−−−−−−−−−→ and flux

4. Distortion5. Time Delay

I The prevalence/interest of one or more of those effectsover the others have to do with the lensing regime.

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

The lensing effect

I Gravitational lensing causes several effects on theimages of the sources:

1. Radial displacement2. Multiple imaging (given certain conditions)3. Magnification of the angular size

surface brightness conservation−−−−−−−−−−−−−−−−−−→ and flux4. Distortion5. Time Delay

I The prevalence/interest of one or more of those effectsover the others have to do with the lensing regime.

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

The lensing effect

I Gravitational lensing causes several effects on theimages of the sources:

1. Radial displacement2. Multiple imaging (given certain conditions)3. Magnification of the angular size

surface brightness conservation−−−−−−−−−−−−−−−−−−→ and flux

4. Distortion5. Time Delay

I The prevalence/interest of one or more of those effectsover the others have to do with the lensing regime.

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

The lensing effect

I Gravitational lensing causes several effects on theimages of the sources:

1. Radial displacement2. Multiple imaging (given certain conditions)3. Magnification of the angular size

surface brightness conservation−−−−−−−−−−−−−−−−−−→ and flux4. Distortion

5. Time Delay

I The prevalence/interest of one or more of those effectsover the others have to do with the lensing regime.

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

The lensing effect

I Gravitational lensing causes several effects on theimages of the sources:

1. Radial displacement2. Multiple imaging (given certain conditions)3. Magnification of the angular size

surface brightness conservation−−−−−−−−−−−−−−−−−−→ and flux4. Distortion5. Time Delay

I The prevalence/interest of one or more of those effectsover the others have to do with the lensing regime.

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

Lensing regimes

I Different lensing regimes differentiate themselves by:

1. The lens mass distribution2. The distances involved3. The impact parameter

I The main lensing regimes are:

1. Micro lensing2. Strong lensing3. Weak lensing

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

Lensing regimes

I Different lensing regimes differentiate themselves by:

1. The lens mass distribution2. The distances involved3. The impact parameter

I The main lensing regimes are:

1. Micro lensing2. Strong lensing3. Weak lensing

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

Micro lensing

1. The lens mass distribution

: Compact, “point mass”

2. The distances involved

: O ∼ tenths of kpc

3. The impact parameter

: variable with very smallminimum

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

Micro lensing

1. The lens mass distribution: Compact, “point mass”

2. The distances involved

: O ∼ tenths of kpc

3. The impact parameter

: variable with very smallminimum

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

Micro lensing

1. The lens mass distribution: Compact, “point mass”

2. The distances involved: O ∼ tenths of kpc

3. The impact parameter

: variable with very smallminimum

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

Micro lensing

1. The lens mass distribution: Compact, “point mass”

2. The distances involved: O ∼ tenths of kpc

3. The impact parameter: variable with very smallminimum

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

The point mass lens

I The lens equation can be easily solved in this case.

I Lets first define a ‘natural scale’, the Einstein angle forthis lens:

θE =

√4GM

c2

Dds

DsDd

I The lens equation then becomes:

β = θ −θ2E

θ

I It has two solutions:

θ± =1

2

(β ±

√β2 + 4θ2

E

)

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

The point mass lens

I The lens equation can be easily solved in this case.

I Lets first define a ‘natural scale’, the Einstein angle forthis lens:

θE =

√4GM

c2

Dds

DsDd

I The lens equation then becomes:

β = θ −θ2E

θ

I It has two solutions:

θ± =1

2

(β ±

√β2 + 4θ2

E

)

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

The point mass lens

I The lens equation can be easily solved in this case.

I Lets first define a ‘natural scale’, the Einstein angle forthis lens:

θE =

√4GM

c2

Dds

DsDd

I The lens equation then becomes:

β = θ −θ2E

θ

I It has two solutions:

θ± =1

2

(β ±

√β2 + 4θ2

E

)

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

The point mass lens

I What is ‘natural’ about the Einstein angle or radius ?

I If β = 0→ θ = θE

I The distance between two images (solutions) is 2θEI The density inside the the Einstein radius is the critical

lensing density

Σcr =c2

4πG

Ds

Dd Dds

above which multiple imaging occur.

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

The point mass lens

I What is ‘natural’ about the Einstein angle or radius ?

I If β = 0→ θ = θE

I The distance between two images (solutions) is 2θEI The density inside the the Einstein radius is the critical

lensing density

Σcr =c2

4πG

Ds

Dd Dds

above which multiple imaging occur.

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

The point mass lens

I What is ‘natural’ about the Einstein angle or radius ?

I If β = 0→ θ = θE

I The distance between two images (solutions) is 2θE

I The density inside the the Einstein radius is the criticallensing density

Σcr =c2

4πG

Ds

Dd Dds

above which multiple imaging occur.

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

The point mass lens

I What is ‘natural’ about the Einstein angle or radius ?

I If β = 0→ θ = θE

I The distance between two images (solutions) is 2θEI The density inside the the Einstein radius is the critical

lensing density

Σcr =c2

4πG

Ds

Dd Dds

above which multiple imaging occur.

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

The point mass lensSolution

θ± =1

2

(β ±

√β2 + 4θ2

E

)x=θ/θE ; y=β/θE−−−−−−−−−−→ x± =

1

2

(y ±

√y + 4)

)Point source

Figura de Narayan & Bartelmann

1996, arXiv:astro-ph/9606001

Extended source

Figura de Narayan & Bartelmann

1996, arXiv:astro-ph/9606001

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

The point mass lensSolution

θ± =1

2

(β ±

√β2 + 4θ2

E

)x=θ/θE ; y=β/θE−−−−−−−−−−→ x± =

1

2

(y ±

√y + 4)

)Point source

Figura de Narayan & Bartelmann

1996, arXiv:astro-ph/9606001

Extended source

Figura de Narayan & Bartelmann

1996, arXiv:astro-ph/9606001

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

Magnification

I The flux magnification qis equal to the ration of theimage and source areas.

I For a circularly symmetric lens:

µ =θ

β

dθ

dβ

I For the point lens, then

µ± =

[1−

(θEθ±

)4]

,

I and

µ = |µ+|+ |µ−| =y2 + 2

y√y2 + 4

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

Magnification

I The flux magnification qis equal to the ration of theimage and source areas.

I For a circularly symmetric lens:

µ =θ

β

dθ

dβ

I For the point lens, then

µ± =

[1−

(θEθ±

)4]

,

I and

µ = |µ+|+ |µ−| =y2 + 2

y√y2 + 4

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

Magnification

I The flux magnification qis equal to the ration of theimage and source areas.

I For a circularly symmetric lens:

µ =θ

β

dθ

dβ

I For the point lens, then

µ± =

[1−

(θEθ±

)4]

,

I and

µ = |µ+|+ |µ−| =y2 + 2

y√y2 + 4

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

Micro lenses

I In nature the micro lenses are usually stars, planets andother compact objects in or close to the Galaxy.

I Due to its proximity the proper motion of those objectsis relevant.

I Figura de Schneider 2006

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

Micro lenses

I In nature the micro lenses are usually stars, planets andother compact objects in or close to the Galaxy.

I Due to its proximity the proper motion of those objectsis relevant.

I Figura de Schneider 2006

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

Micro lenses

I Typical Einstein radius

θE = 0.902mas

(M

M�

)1/2( Dd

10kpc

)−1/2(1− Dd

Ds

)−1/2

I We cannot observationally resolve the multiple imaging,but we can measure the variation of the flux due to themagnification

I Note that microlensing should produce symmetric andachromatic light curves as above.

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

Micro lenses

I Typical Einstein radius

θE = 0.902mas

(M

M�

)1/2( Dd

10kpc

)−1/2(1− Dd

Ds

)−1/2

I We cannot observationally resolve the multiple imaging,but we can measure the variation of the flux due to themagnification

I Note that microlensing should produce symmetric andachromatic light curves as above.

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

Micro lenses

I Typical Einstein radius

θE = 0.902mas

(M

M�

)1/2( Dd

10kpc

)−1/2(1− Dd

Ds

)−1/2

I We cannot observationally resolve the multiple imaging,but we can measure the variation of the flux due to themagnification

I Note that microlensing should produce symmetric andachromatic light curves as above.

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

Astrophysical Applications of Microlensing

1. Extra solar planet detection

OGLE-2005-BLG-390; Credit:ESO

2. Stellar & Galactic astrophysics

3. Quantification of the MAssive Compact Halo Objects(MACHOs) in the Galaxy.

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

Astrophysical Applications of Microlensing

1. Extra solar planet detection

OGLE-2005-BLG-390; Credit:ESO

2. Stellar & Galactic astrophysics

3. Quantification of the MAssive Compact Halo Objects(MACHOs) in the Galaxy.

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

Astrophysical Applications of Microlensing

1. Extra solar planet detection

OGLE-2005-BLG-390; Credit:ESO

2. Stellar & Galactic astrophysics

3. Quantification of the MAssive Compact Halo Objects(MACHOs) in the Galaxy.

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

MACHOs in the Galaxy

I Two main projects: MACHO & EROS: Observation ofthe LMC & SMC

Figure from Alcock et al. 2000

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

MACHOs in the Galaxy

I Macho project: 5.7 years; LMC.

→ 13-17 events ; 100%MACHO halo ruled out with 95% c.l.

I EROS project: 5 years; SMC

→ 5 events; 25% of less ofthe MW halo is composed by MACHOs (95% c.l.)

Figure from Afonso et al. 2003

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

MACHOs in the Galaxy

I Macho project: 5.7 years; LMC. → 13-17 events ; 100%MACHO halo ruled out with 95% c.l.

I EROS project: 5 years; SMC

→ 5 events; 25% of less ofthe MW halo is composed by MACHOs (95% c.l.)

Figure from Afonso et al. 2003

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

MACHOs in the Galaxy

I Macho project: 5.7 years; LMC. → 13-17 events ; 100%MACHO halo ruled out with 95% c.l.

I EROS project: 5 years; SMC

→ 5 events; 25% of less ofthe MW halo is composed by MACHOs (95% c.l.)

Figure from Afonso et al. 2003

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

MACHOs in the Galaxy

I Macho project: 5.7 years; LMC. → 13-17 events ; 100%MACHO halo ruled out with 95% c.l.

I EROS project: 5 years; SMC → 5 events; 25% of less ofthe MW halo is composed by MACHOs (95% c.l.)

Figure from Afonso et al. 2003

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

MACHOs in the Galaxy

Microlensing surveys, in practice, consolidate the notion thatdark matter must be constituted by some kind of (non-baryonic) particles, instead of microscopic objects

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

Strong lensing

1. The lens mass distribution

: extended - galaxies andgalaxy clusters

2. The distances involved

: O ∼ Gpc

3. The impact parameter

: ≤ θE → multiple imaging

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

Strong lensing

1. The lens mass distribution: extended - galaxies andgalaxy clusters

2. The distances involved

: O ∼ Gpc

3. The impact parameter

: ≤ θE → multiple imaging

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

Strong lensing

1. The lens mass distribution: extended - galaxies andgalaxy clusters

2. The distances involved: O ∼ Gpc

3. The impact parameter

: ≤ θE → multiple imaging

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

Strong lensing

1. The lens mass distribution: extended - galaxies andgalaxy clusters

2. The distances involved: O ∼ Gpc

3. The impact parameter: ≤ θE → multiple imaging

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

Extendend lenses

I In practice the lens dimensions are much smaller thanthe other distances involved.

I In this limit the thin lens approximation is valid:

The relevant quantity is not the 3D mass distribution(ρ) but instead the projected (2D) mass density (Σ)

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

Extendend lenses

I In practice the lens dimensions are much smaller thanthe other distances involved.

I In this limit the thin lens approximation is valid:

The relevant quantity is not the 3D mass distribution(ρ) but instead the projected (2D) mass density (Σ)

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

Extendend lenses

I In practice the lens dimensions are much smaller thanthe other distances involved.

I In this limit the thin lens approximation is valid:The relevant quantity is not the 3D mass distribution(ρ) but instead the projected (2D) mass density (Σ)

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

Extendend lenses

I The deflection angle can then be calculated as the sumofthe contributions of every mass element:

α =∑i

4 G mi

c2

ξ − ξi|ξ − ξi |2

I For a continuous distribution mass distribution:

α(ξ) =4G

c2

∫Σ(ξ′)

ξ − ξ′

|ξ − ξ′|2d2ξ′

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

Extendend lenses

I The deflection angle can then be calculated as the sumofthe contributions of every mass element:

α =∑i

4 G mi

c2

ξ − ξi|ξ − ξi |2

I For a continuous distribution mass distribution:

α(ξ) =4G

c2

∫Σ(ξ′)

ξ − ξ′

|ξ − ξ′|2d2ξ′

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

Lens models

I Singular Isothermal Sphere (SIS)

ρ(r) =σ2v

2πGr2

- Elliptical galaxies

I Navarro, Frenk & White (NFW)

ρ(r) =ρs

(r/rs)(1 + r/rs)2

- Clusters of Galaxies

ρ(r < rs) ∝ r−1; ρ(r ∼ rs) ∝ r−2; ρ(r > rs) ∝ r−3

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

Lens models

I Singular Isothermal Sphere (SIS)

ρ(r) =σ2v

2πGr2

- Elliptical galaxies

I Navarro, Frenk & White (NFW)

ρ(r) =ρs

(r/rs)(1 + r/rs)2

- Clusters of Galaxies

ρ(r < rs) ∝ r−1; ρ(r ∼ rs) ∝ r−2; ρ(r > rs) ∝ r−3

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

Lens models

I Singular Isothermal Sphere (SIS)

ρ(r) =σ2v

2πGr2

- Elliptical galaxies

I Navarro, Frenk & White (NFW)

ρ(r) =ρs

(r/rs)(1 + r/rs)2

- Clusters of Galaxies

ρ(r < rs) ∝ r−1; ρ(r ∼ rs) ∝ r−2; ρ(r > rs) ∝ r−3

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

Lens models

I Singular Isothermal Sphere (SIS)

ρ(r) =σ2v

2πGr2

- Elliptical galaxies

I Navarro, Frenk & White (NFW)

ρ(r) =ρs

(r/rs)(1 + r/rs)2

- Clusters of Galaxies

ρ(r < rs) ∝ r−1; ρ(r ∼ rs) ∝ r−2; ρ(r > rs) ∝ r−3

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

Lens models

I Singular Isothermal Sphere (SIS)

ρ(r) =σ2v

2πGr2

- Elliptical galaxies

I Navarro, Frenk & White (NFW)

ρ(r) =ρs

(r/rs)(1 + r/rs)2

- Clusters of Galaxies

ρ(r < rs) ∝ r−1; ρ(r ∼ rs) ∝ r−2; ρ(r > rs) ∝ r−3

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

Strong Lensing ModellingI Given a lens mass model and the position of the images

we can solve the lens equation

β = θ −α(θ)

I This is a mapping problem:

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

Strong Lensing ModellingI Given a lens mass model and the position of the images

we can solve the lens equation

β = θ −α(θ)

I This is a mapping problem:

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

Strong Lensing ModellingCircular Lens

Source Plane Image Plane

Figure from Hattori, Kneib & Makino, arXiv:astro-ph/9905009

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

Strong Lensing ModellingCircular Lens

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

Strong Lensing ModellingElliptical Lens

Source Plane Image Plane

Figure from Hattori, Kneib & Makino, arXiv:astro-ph/9905009

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

Strong Lensing ModellingElliptical Lens

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

Giants arcs in Galaxy ClustersResults

1. Reinforcement that clusters are dark matter dominated(M/L ∼ 100− 300M�/L�)

2. Arcs are large and relatively regular

→ The dark matteris smoothly distributed instead of attached to galaxies.

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

Giants arcs in Galaxy ClustersResults

1. Reinforcement that clusters are dark matter dominated(M/L ∼ 100− 300M�/L�)

2. Arcs are large and relatively regular

→ The dark matteris smoothly distributed instead of attached to galaxies.

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

Giants arcs in Galaxy ClustersResults

1. Reinforcement that clusters are dark matter dominated(M/L ∼ 100− 300M�/L�)

2. Arcs are large and relatively regular → The dark matteris smoothly distributed instead of attached to galaxies.

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

Giants arcs in Galaxy ClustersResults

1. Reinforcement that clusters are dark matter dominated(M/L ∼ 100− 300M�/L�).

2. The dark matter is smoothly distributed instead ofattached to galaxies.

3. Strong lensing clusters are not symmetrical and oftensubstructured.

4. The slope of the mass profile at the very centre isshallower than numerical predictions.

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

Giants arcs in Galaxy ClustersResults

1. Reinforcement that clusters are dark matter dominated(M/L ∼ 100− 300M�/L�).

2. The dark matter is smoothly distributed instead ofattached to galaxies.

3. Strong lensing clusters are not symmetrical and oftensubstructured.

4. The slope of the mass profile at the very centre isshallower than numerical predictions.

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

Giants arcs in Galaxy ClustersResults

1. Reinforcement that clusters are dark matter dominated(M/L ∼ 100− 300M�/L�).

2. The dark matter is smoothly distributed instead ofattached to galaxies.

3. Strong lensing clusters are not symmetrical and oftensubstructured.

4. The slope of the mass profile at the very centre isshallower than numerical predictions.

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

Galaxy strong lensing

Figure from Bolton et al. (2008)

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

Galaxy strong lensingResults for early-type galaxies

1. Dark matter is about 1/4 of the mass at the half-lightradius (70% at 5 re).

2. The mass profile is isothermal (ρ ∝ r−2) with very littlescatter.

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

Galaxy strong lensingResults for early-type galaxies

1. Dark matter is about 1/4 of the mass at the half-lightradius (70% at 5 re).

2. The mass profile is isothermal (ρ ∝ r−2) with very littlescatter.

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

Weak lensing

1. The lens mass distribution

: extended - galaxies, clustersand the large scale structure

2. The distances involved

: O ∼ Gpc

3. The impact parameter

: � θE

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

Weak lensing

1. The lens mass distribution: extended - galaxies, clustersand the large scale structure

2. The distances involved

: O ∼ Gpc

3. The impact parameter

: � θE

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

Weak lensing

1. The lens mass distribution: extended - galaxies, clustersand the large scale structure

2. The distances involved: O ∼ Gpc

3. The impact parameter

: � θE

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

Weak lensing

1. The lens mass distribution: extended - galaxies, clustersand the large scale structure

2. The distances involved: O ∼ Gpc

3. The impact parameter: � θE

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

Gravitational Image Distortion

I The gravitational lensing distorts the images

I The isotropic distortion is associated with theconvergence: κ = Σ(r)/Σcr

and anisotropic with the shear: γt × Σcr = Σ(r)− Σ(r)

I Both κ and γ are second derivatives of the lensingeffective potential.

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

Gravitational Image Distortion

I The gravitational lensing distorts the images

I The isotropic distortion is associated with theconvergence: κ = Σ(r)/Σcr

and anisotropic with the shear: γt × Σcr = Σ(r)− Σ(r)

I Both κ and γ are second derivatives of the lensingeffective potential.

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

Gravitational Image Distortion

I The gravitational lensing distorts the images

I The isotropic distortion is associated with theconvergence: κ = Σ(r)/Σcr

and anisotropic with the shear: γt × Σcr = Σ(r)− Σ(r)

I Both κ and γ are second derivatives of the lensingeffective potential.

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

I Given the large impact parameter the distortion is verysmall and is dominated by the galaxies own shapes.

I The weak shear can be detected statistically, however,by using samples with large numbers of galaxies:〈e〉 = γ/(1− κ) ' γ

I Allow mass measurements far beyond the strong lensingarea.

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

I Given the large impact parameter the distortion is verysmall and is dominated by the galaxies own shapes.

I The weak shear can be detected statistically, however,by using samples with large numbers of galaxies:〈e〉 = γ/(1− κ) ' γ

I Allow mass measurements far beyond the strong lensingarea.

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

I Given the large impact parameter the distortion is verysmall and is dominated by the galaxies own shapes.

I The weak shear can be detected statistically, however,by using samples with large numbers of galaxies:〈e〉 = γ/(1− κ) ' γ

I Allow mass measurements far beyond the strong lensingarea.

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

I Weak-lensing measurements are quite delicate as onehas to measure shapes of small source galaxies

Figure from Bridle et al. (2008; arXiv:0802.1214)

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

Weak lensing mass mapsI The grav. shear field can be used to reconstruct the

projected mass distribution

Figure from Richard Massey (CalTech)

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

The Bullet Cluster

I On the merging cluster 1E 0657-56 gravity does notfollow the baryonic mass (mostly gas)

One of the most convincing evidences of the existence of darkmatter.

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

Cluster Mass Profiles

I General high quality results can be obtaining bystacking the weak lensing signal of samples of lenses.

Figure from Okabe et al. (2010)

I The cluster mass profile is curved in a log-log space andconsistent with numerical simulations such as NFW.

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

Cluster Mass Profiles

I General high quality results can be obtaining bystacking the weak lensing signal of samples of lenses.

Figure from Okabe et al. (2010)

I The cluster mass profile is curved in a log-log space andconsistent with numerical simulations such as NFW.

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

Mass measurementsI Weak grav. lensing cluster mass measurements are

regarded as the most trustworthy as they do not sufferfrom the hydrostatic mass bias.

I They are ideal to calibrate mass-observable relations.

Figure from Mantz et al. (2016; arXiv:1606.03407)

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

Mass measurements

I Those scale relations are crucial for cluster-count basedcosmology. Constrains in the (dark) matter density andclumpiness

Figure from Mantz et al. (2014)

I With a larger redshift range constrains on dark energyas well.

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

Mass measurements

I Those scale relations are crucial for cluster-count basedcosmology. Constrains in the (dark) matter density andclumpiness

Figure from Mantz et al. (2014)

I With a larger redshift range constrains on dark energyas well.

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

Cosmic shear

I Weak lensing by the large scale structure.

Figures from http://www.cfht.hawaii.edu/News/Lensing/

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

Cosmic shear

I The signal is even smaller than the cluster weak lensing

I Measurements uses the 2-point correlation of shapes.

Figure from Kaiser et al. (2000; arXiv:astro-ph/0003338

A primer onGravitational

Lensing

Eduardo S.Cypriano

GravitationalLensing basics

Gravitational lensinggeometry

The lens equation

The deflection angle

Micro lensing

The point mass lens

Magnification

Micro lenses

AstrophysicalApplications ofMicrolensing

Strong lensing

Extendend lenses

Lens models

Strong LensingModelling

Main Results

Weak Lensing

Gravitational ImageDistortion

The Mass Distributionin Clusters

Cosmic shear

Cosmic shear

I Similar sensitive to cosmological parameters ascluster-counts.

Figure from Heymans et al. (2013)

I Cosmics shear and Cluster counts have the potential todiscriminate between dark energy and modifiedgravitation.